On the sign of solutions to some linear parabolic biharmonic equations
The author studies the sign of the solution of the following Cauchy problem \[ \begin{cases} u_t+\Delta ^2u=g(x,t) &\text{in }\mathbb R^n\times[0,\infty), \\ u(x,0)=u_0(x) &\text{in }\mathbb R^n, \end{cases} \] where \(n\) is a positive integer, \(g\) and \(u_0\) are continuous functions. The first purpose of the paper is to study which assumptions on the source \(g \) guarantee eventual local positivity for the solution. The author starts by studying the case of sources \(g\) and data \(u_0\) having compact support. Then the author relaxes this assumption and deals with sources having compact support only with respect to the time or to the space variable. Finally, the assumption on the support of \(g\) is replaced by suitable decay conditions with respect to both space and time. The second purpose of the paper is to discuss the problem of getting global positivity. When the source \(g\) has compact support with respect to one or both of its variable, the author shows that negativity may always occur, thus, in general, no globally positivity can be expected. Furthermore some examples where global positivity fails are given.
scientific article; zbMATH DE number 5573015